Ergodic Hypothesis
In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a particle in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time. A long period of time will depend on the time constants of the processes involved and the number of different components, both initial and possible. Use caution in assuming ergodic performance when the number of components, N, is large; it is likely that the time to completion will vary as N2. For complex biological systems, ergodic performance is seldom possible. Ludwig Boltzmann accomplished the feat of showing that the second law of thermodynamics is only a statistical fact. The gradual disordering of energy is analogous to the disordering of an initially ordered pack of cards under repeated shuffling, and just as the cards will finally return to their original order if shuffled a gigantic number of times, so the entire universe must some day regain, by pure chance, the state from which it first set out. (This optimistic coda to the idea of the dying universe becomes somewhat muted when one attempts to estimate the timeline which will probably elapse before it spontaneously occurs.)"Collier's Encyclopedia", Volume 19 Phyfe to Reni, Physics, by David Park, p. 15 The tendency for entropy increase seems to cause difficulty to beginners in thermodynamics, but is easy to understand from the standpoint of the theory of probability. Consider two ordinary dice, with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small (1 in 36); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the system must move to one of the more probable states."Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275 However, mathematically the odds of all the dice results not being a pair sixes is also as hard as the ones of all of them being sixes, and since statistically the data tend to balance, one in every 36 pairs of dice will tend to be a pair of sixes. And the cards, when shuffled, will sometimes present a certain temporary sequence order even if in its whole they are disordered. The ergodic hypothesis is often assumed in statistical analysis. The analyst would assume that the average of a process parameter over time and the average over the statistical ensemble are the same. Right or not, the analyst assumes that it is as good to observe a process for a long time as sampling many independent realisations of the same process. Liouville's Theorem shows that, for conserved classical systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the total or convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase space initially, they will remain so at all times. Liouville's theorem ensures that the notion of time average makes sense, but ergodicity does not follow from Liouville's theorem. Phenomenology In macroscopic systems, the timescales over which a system can truly explore the entirety of its own phase space can be sufficiently large that the thermodynamic equilibrium state exhibits some form of ergodicity breaking. A common example is that of spontaneous magnetisation in ferromagnetic systems, whereby below the Curie temperature the system preferentially adopts a non-zero magnetisation even though the ergodic hypothesis would imply that no net magnetisation should exist by virtue of the system exploring all states whose time-averaged magnetisation should be zero. The fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking. However, complex disordered systems such as a spin glass show an even more complicated form of ergodicity breaking where the properties of the thermodynamic equilibrium state seen in practice are much more difficult to predict purely by symmetry arguments. Also conventional glasses (e.g. window glasses) violate ergodicity in a complicated manner. In praxis this means that on sufficiently short time scales (e.g. those of parts of seconds, minutes, or a few hours) the systems may behave as solids, i.e. with a positive shear modulus, but on extremely long scales, e.g. in millennia or eons, as liquids, or with two or more time scales and plateaux in between.The introduction of the practical aspect of ergodicity breaking by introducing a "non-ergodicity time scale" is due to R.G. Palmer, Adv. Phys. 31, 669 (1982). Also related to these time-scale phenomena are the properties of ageing and the Mode-Coupling theory of W. Götze, Dynamics of Glass Forming Liquids, Oxford Univ. Press, 2008 Mathematics Ergodic theory is a branch of mathematics which deals with dynamical systems that satisfy a version of this hypothesis, phrased in the language of measure theory. See also * Poincaré recurrence theorem * Loschmidt's paradox * Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicity * Ergodic process * Ergodicity References Category:Ergodic theory Category:Statistical mechanics Category:Philosophy of thermal and statistical physics Category:Fundamental physics concepts